Mixing
How do I prove the equivalence between surjectivity or injectivity and existence of one sided inverse...
$begingroup$
I am having a problem with a three piece exercise.
It goes like this:
a) Show that there exists a function $g$ such that $fcirc g = I$ if and only if $f$ is surjective.
b) Show that there exists a function $h$ such that $hcirc f = I$ if and only if $f$ is injective.
c) If $fcirc g = fcirc h = I$, is it always true that $g = h$?
Here $I$ is the identity function and all functions go from $mathbb{R}$ to $mathbb{R}$.
I think the first two make perfect sense but I have no idea how to effectively show the implications towards the existence of a function.
Thank you in advance for all the help.
abstract-algebra
asked Jan 30 at 17:14
DaàvidDaàvid
505
$endgroup$
|
show 2 more comments
$begingroup$
I am having a problem with a three piece exercise.
It goes like this:
a) Show that there exists a function $g$ such that $fcirc g = I$ if and only if $f$ is surjective.
b) Show that there exists a function $h$ such that $hcirc f = I$ if and only if $f$ is injective.
c) If $fcirc g = fcirc h = I$, is it always true that $g = h$?
Here $I$ is the identity function and all functions go from $mathbb{R}$ to $mathbb{R}$.
I think the first two make perfect sense but I have no idea how to effectively show the implications towards the existence of a function.
Thank you in advance for all the help.
abstract-algebra
asked Jan 30 at 17:14
DaàvidDaàvid
505
$endgroup$
$begingroup$
Try to go step by step, one implication at a time, working from/towards the definitions. So for part a, start by assuming such g exists. Now use that g to show surjectivity, ie take an arbitrary y and write down x such that f(x)=y.
$endgroup$
– Nathaniel Mayer
Jan 30 at 17:43
$begingroup$
I really am not seeing how to do this. Maybe I don’t know all the definitions, but I really don’t even know what would be missing
$endgroup$
– Daàvid
Jan 30 at 18:13
$begingroup$
I just gave you the definition of surjectivity. Suppose you have functions f and g and a real number y, and that f(g) = I. Find x such that f(x)=y. Does that make sense?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:15
$begingroup$
If you can't get the formality, maybe start by writing in sentences why it "makes perfect sense"
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:16
$begingroup$
Oh, thank you. I knew those definitions, I just was taking the functions in the opposite order and it wasn’t making sense
$endgroup$
– Daàvid
Jan 30 at 18:18
|
show 2 more comments
$begingroup$
I am having a problem with a three piece exercise.
It goes like this:
a) Show that there exists a function $g$ such that $fcirc g = I$ if and only if $f$ is surjective.
b) Show that there exists a function $h$ such that $hcirc f = I$ if and only if $f$ is injective.
c) If $fcirc g = fcirc h = I$, is it always true that $g = h$?
Here $I$ is the identity function and all functions go from $mathbb{R}$ to $mathbb{R}$.
I think the first two make perfect sense but I have no idea how to effectively show the implications towards the existence of a function.
Thank you in advance for all the help.
abstract-algebra
asked Jan 30 at 17:14
DaàvidDaàvid
505
$endgroup$
I am having a problem with a three piece exercise.
It goes like this:
a) Show that there exists a function $g$ such that $fcirc g = I$ if and only if $f$ is surjective.
b) Show that there exists a function $h$ such that $hcirc f = I$ if and only if $f$ is injective.
c) If $fcirc g = fcirc h = I$, is it always true that $g = h$?
Here $I$ is the identity function and all functions go from $mathbb{R}$ to $mathbb{R}$.
I think the first two make perfect sense but I have no idea how to effectively show the implications towards the existence of a function.
Thank you in advance for all the help.
abstract-algebra
abstract-algebra
asked Jan 30 at 17:14
DaàvidDaàvid
505
asked Jan 30 at 17:14
DaàvidDaàvid
505
edited Feb 1 at 7:47
Daàvid
asked Jan 30 at 17:14
DaàvidDaàvid
505
asked Jan 30 at 17:14
DaàvidDaàvid
505
asked Jan 30 at 17:14
DaàvidDaàvid
505
505
$begingroup$
Try to go step by step, one implication at a time, working from/towards the definitions. So for part a, start by assuming such g exists. Now use that g to show surjectivity, ie take an arbitrary y and write down x such that f(x)=y.
$endgroup$
– Nathaniel Mayer
Jan 30 at 17:43
$begingroup$
I really am not seeing how to do this. Maybe I don’t know all the definitions, but I really don’t even know what would be missing
$endgroup$
– Daàvid
Jan 30 at 18:13
$begingroup$
I just gave you the definition of surjectivity. Suppose you have functions f and g and a real number y, and that f(g) = I. Find x such that f(x)=y. Does that make sense?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:15
$begingroup$
If you can't get the formality, maybe start by writing in sentences why it "makes perfect sense"
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:16
$begingroup$
Oh, thank you. I knew those definitions, I just was taking the functions in the opposite order and it wasn’t making sense
$endgroup$
– Daàvid
Jan 30 at 18:18
|
show 2 more comments
$begingroup$
Try to go step by step, one implication at a time, working from/towards the definitions. So for part a, start by assuming such g exists. Now use that g to show surjectivity, ie take an arbitrary y and write down x such that f(x)=y.
$endgroup$
– Nathaniel Mayer
Jan 30 at 17:43
$begingroup$
I really am not seeing how to do this. Maybe I don’t know all the definitions, but I really don’t even know what would be missing
$endgroup$
– Daàvid
Jan 30 at 18:13
$begingroup$
I just gave you the definition of surjectivity. Suppose you have functions f and g and a real number y, and that f(g) = I. Find x such that f(x)=y. Does that make sense?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:15
$begingroup$
If you can't get the formality, maybe start by writing in sentences why it "makes perfect sense"
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:16
$begingroup$
Oh, thank you. I knew those definitions, I just was taking the functions in the opposite order and it wasn’t making sense
$endgroup$
– Daàvid
Jan 30 at 18:18
$begingroup$
Try to go step by step, one implication at a time, working from/towards the definitions. So for part a, start by assuming such g exists. Now use that g to show surjectivity, ie take an arbitrary y and write down x such that f(x)=y.
$endgroup$
– Nathaniel Mayer
Jan 30 at 17:43
$begingroup$
Try to go step by step, one implication at a time, working from/towards the definitions. So for part a, start by assuming such g exists. Now use that g to show surjectivity, ie take an arbitrary y and write down x such that f(x)=y.
$endgroup$
– Nathaniel Mayer
Jan 30 at 17:43
$begingroup$
I really am not seeing how to do this. Maybe I don’t know all the definitions, but I really don’t even know what would be missing
$endgroup$
– Daàvid
Jan 30 at 18:13
$begingroup$
I really am not seeing how to do this. Maybe I don’t know all the definitions, but I really don’t even know what would be missing
$endgroup$
– Daàvid
Jan 30 at 18:13
$begingroup$
I just gave you the definition of surjectivity. Suppose you have functions f and g and a real number y, and that f(g) = I. Find x such that f(x)=y. Does that make sense?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:15
$begingroup$
I just gave you the definition of surjectivity. Suppose you have functions f and g and a real number y, and that f(g) = I. Find x such that f(x)=y. Does that make sense?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:15
$begingroup$
If you can't get the formality, maybe start by writing in sentences why it "makes perfect sense"
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:16
$begingroup$
If you can't get the formality, maybe start by writing in sentences why it "makes perfect sense"
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:16
$begingroup$
Oh, thank you. I knew those definitions, I just was taking the functions in the opposite order and it wasn’t making sense
$endgroup$
– Daàvid
Jan 30 at 18:18
$begingroup$
Oh, thank you. I knew those definitions, I just was taking the functions in the opposite order and it wasn’t making sense
$endgroup$
– Daàvid
Jan 30 at 18:18
|
show 2 more comments
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$begingroup$
Try to go step by step, one implication at a time, working from/towards the definitions. So for part a, start by assuming such g exists. Now use that g to show surjectivity, ie take an arbitrary y and write down x such that f(x)=y.
$endgroup$
– Nathaniel Mayer
Jan 30 at 17:43
$begingroup$
I really am not seeing how to do this. Maybe I don’t know all the definitions, but I really don’t even know what would be missing
$endgroup$
– Daàvid
Jan 30 at 18:13
$begingroup$
I just gave you the definition of surjectivity. Suppose you have functions f and g and a real number y, and that f(g) = I. Find x such that f(x)=y. Does that make sense?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:15
$begingroup$
If you can't get the formality, maybe start by writing in sentences why it "makes perfect sense"
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:16
$begingroup$
Oh, thank you. I knew those definitions, I just was taking the functions in the opposite order and it wasn’t making sense
$endgroup$
– Daàvid
Jan 30 at 18:18